In the present paper, we analytically studied the properties of
the holographic \(s\) -wave superconductors in the presence of
exponential correction to the Maxwell electrodynamics in the
background of Schwarzschild AdS black holes. Adopting the probe
limit, where the scalar and gauge field do not back react on the
metric background, we found the nonlinear coupling parameter
influences the critical temperature and the condensation operator
near the critical point. We performed our calculations up to the
first order in the nonlinear coupling parameter \(b\) . We observed
that the critical temperature \(T_{c}\)  decreases with increasing
the nonlinear parameter \(b\) , which means that the condensation
become harder to be formed for higher nonlinear corrections. This
result is similar to the holographic \(s\) -wave superconductors with
BI nonlinear electrodynamics [1]}. However, we found that
the EN electrodynamics has stronger effect on the condensation
formation compared to the BI electrodynamics. The analytical
results obtained in this work are in good agreement with the
existing numerical results for the holographic superconductors
with EN electrodynamics [2]}. The critical exponent of
the condensation also comes out to be \(1/2\)  which is the universal
value in the mean field theory. Thus, the superconductor phase
transition appeared in the linear Maxwell electrodynamics also
exists in its nonlinear extension. Our work helps to give a better
understanding of the holographic superconductors when the gauge
field is in the form of EN electrodynamics.
